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"0! ""61!13 424!2 HFK 8!99224B!034 2 !412034 2 ; 424!2< 1 2@24!2< 85L22 /22: 4/034D 1 / (t0 , y0) = (0, 2) 6 ?21 !6213 14I24" I0 40 4< I4!/244.7 "2 /!"0 6 −1< 2@24!2< ; 85L22 /22: 1 / (0, 1) 6 ?21 !6213 14I24" I0 40 4< I4!/244.7 "4!: :4/24!26 −2.
1!6034 2 0D/24!2 Y (t), Y (t) 1 /4 I 2@24!5 y 8 021 521 "!; "1262
Y = −Y , Y (0) = 2 Y = −Y , Y (0) = 1
028 12034 <
y(t) ∈ Y (t), Y (t) = (3e−t − et )/2, (3e−t + et )/2 ,
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√ √ √ √ ⎤ √ √ √ √ 1 − 2 2 2 2 1 + −1 + 1 + ⎣ √ e 2t + e− 2t , e 2t + √ e− 2t ⎦ , t ∈ [0, 0.623]. 2 2 2 2 2 2 ⎡
""61!5 2@24!2 621!: 44 2 "2627"1 < 6. 6 ?26 0; /!13 I4!B e−2t, e−t . A 1< "1 5L! !412034.2 621 8.< !62D1 62"1 A C1! 8 8< 15 805 4B2B!!< 1 7 6. " "10526 "!"1262 A.4 244. 8!9; 9224B!034. 424!7 621!: 44 2 "2627"1 < :4!D1 "0 ?4 ; "1! " 4 ?824!26 C9921!4. /!"0244. 0I !16 & !62< 2"0! :8/2 H$K 0D/24. 621. 94B!! f +N "/!121< /1 ?8.7 621-C1 8 04!120345 :!"!65 2262445< !: 845 1 7 4 40D& ?8.7 !: 621 8 B24! I0 A034 7 @!A!< !"44.7 A 12 MG#N 6 ?21 A: .13 A:!" !412034 I 0I !16& ?824!2 0D/24!7 805 y(t) "4 4 4 • 8!99224B!034. 4224"1< • 42/4 ;:4 "14. "!6B!5 " B24 7 I2@4 "1!<
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4B!5 0D/24!5 6 ?21 A.13 " 4"1! 4 !: !412 0541 < !"; 03:26. C1!6! 621 86!& 412034.2 621 8.< "4 44.2 4 42/4 -:4 "14. "!6; B!5< 42 505D1"5 C9921!4.6!< " 03 4! 8D1 0D/24!5 0!@3 :0 "21!< 42 94B!! 0D/24!5< ! 1 6< /1 621 8. 58 270 8D1 A./4 A 022 :!2 0D/24!5& 60 !8 yi+1 = yi +. . . , ./!"05265 " !" 03: 4!26 !412034 7 !9621!!< ! 8!1 5024!D !412034. 94B!7< @!!4 1 ; . :"121 " 20!/24!26 :4/24!7 I6241 H :"1DL! @!; !42K& 84 . 9 60< !" 03:26. 64 I!6! !:2"14.6! 0I !16; 6! 805 41 05 : 20!/!4 7 @I< 505D1"5 !0212034.6! 805 !4; 12034 I !" 0424!5& !412034 7 "14 2 :4/24!5 !4120 H0D/24!5K< 0/244.2 86 !06 "02821 22"213< /1 A. 4 ; 8!13 A 022 : 2 0D/24!2& " ! A 1 M$< FN !" 03: 0 !9621! 8!99224B!0 < /1 ; A. " :813 58. 270 805 2@24!7 ! !412034.2 621 8.< A2"2/!; DL!2 0D/24!2 "11 /4 I /024 58 & "13 Ti = [ti , ti+1]< yˆi A :4/21 B24 2@24!5 y(ti ) = yi . :0I0 yˆ(t) 58 270 1 /2 t = ti " @I 6 h = t − ti : (p)
yˆ(t) = yˆi + yˆi h + yˆi h2 /2! + . . . + yˆi hp /p! + ep
!: 84.2 yˆi(j) = yˆj (ti ) ./!"05D1"5 !: 8!99224B!034 I 424!5 " 6 L3D !9621!! 8!99224B!0 ep = yˆ(p+1) (ξ)hp+1/(p + 1)!, ξ ∈ Ti
A./4 2 !62424!2 621 8 58 270 " 62"1!6 " "14814.6! 64 I @I .6! 621 86! 1! 4I2;11 805 2@24!5 42?2"1! :8/ 6 A./4 7 H42!412034 7 !9621!!K&
"0! Yi - !4120< 1 7 /1 yi ∈ Yi < 1 y(t) ∈ Yi (t) 805 t ∈ Ti < I82 (p)
Yi+1(t) = Yi + Yi h + Yi h2 /2! + . . . + Yi hp /p! + Ep .
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424!! H'K Yi(j) ./!"050 "3 " !" 03: 4!26 1 /4 I 22414 ; I " 14 @24!5 1 I < 805 yˆi(j) " 162/244.6! 2B!56!< 20!: 4; 4.6! !412034 7 !9621!2& @!A "2/24!5 :0D/24 !4120 (p+1)
E p = Yi
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:6 ?4. 64 I!2 !41. 621 8 2B!! 82921< 84 "2 4! !62D1 "028DLD ALD 9 6S • ./!"0521"5 "!6B!5 2@24!5 yˆ(t) 805 y(t)< • 82921 2820521"5 d(t) = yˆ − f (t, yˆ(t))< • !" 03:21"5 2@24!5 yˆ :8/!< C99!B!241. 1 7 A0!:! C9; 9!B!2416 !" 84 7 :8/!< yˆ = f (t, y)+d(t) H"L2"1 4!2 2@24!5K< • ./!"0521"5 yˆˆ- !A0!?244 2 2@24!2 :8/! y = f (t, y) + d(t) 1 6 ?2 0I !16< /1 ! yˆ(t),< • 28 0I21"5< /1 yˆˆ − yˆ(t) ≈ yˆ(t) − y(t), 1 2"13 0I26< /1
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42412 AP28!424!2 !412034 I "@!24!5 "074;94B!!< "!6!DL27 2@24!2 !" 84 7 "!"126. A.4 244. 8!992; 24B!034. 424!7 ! !412034 7 B24! I0 A034 7 @!A!
AB!5 A 1.& 2:0311. 8!""21B!! 8 08.0!"3 4 "26!42 928. ./!"0!12034. 621 8 624!! "0 @4 7 "28. "!A!" I I "8"1244 I 4!2"!121 8 8"1 6 ; 826! && 424 \ 4 "26!42 /!"0244 6 40!: 8 8"1 6 826! && !4 H < I& "!A!"K\ 4 "26!42 m A026. 61261!/2" I ! /!"0244 I 6 820! 4!5n H I& "4 5"K\ 4 49224B!5 6 0 8. /24. I& "!A!" H$%J$< $%JG II&K\
4 62?84 84 6 "!6 :!62 !412034 7 61261!2 H 57AI< 264!5< $%J+ I&K\ 4 "2 ""!7"! " 2L4!5 !412034 6 40!: H $%J#< $%J'< $%JJII& o I& "4 5"< $%%FI& o A;D" K\ 4 "2 ""!7" 7 @ 02 m ./!"0!12034.2 621 8. ! 61261!/2" 2 6 82; 0! 4!2n - 4"1!11 !084 7 61261!! !6&& &208.@< "; 4 5"!7 I "4!2"!121 H$%J' I& - @24" 2K\ 4 "2 ""!7" 7 49224B!! m1034.2 A026. !084 7 6126; 1!!n - 1 "!7 I "4!2"!121 H $%%$I& - 1 K\ 4 J;7 62?84 84 7 @ 02;"26!42 m/2"12445 12 !5 8!99224; B!034. 424!7 I!8 8!46!!n o 4"1!11 I!8 8!46!! !6& &&(24132< "4 5"!7 I "4!2"!121 H $%%FI& - "4 5"K\ 4 62?84 84 7 49224B!! /!"0244 6 40!: ! 1 61!/2" 7 2!9!B!! 2:0311 - 4!2"!121 @11 845 (!:!4 H$%%GI& - (!:!4< K\ 4 62?84 84 7 49224B!! 63D124.6 "!"1266 ! !084 7 61261!2 - &;212AI"!7 I "4!2"!121 H$%%GI& - &;212AIK\ 4 62?84 84 7 49224B!! !412034.6 ! 63D124 ;0I2A!/2"!6 621 86 42 ! !4?242!! - &;212AI"!7 I "4!2"!121 H $%%>I& o &;212AIK\ 4 "2 ""!7" 6 " 2L4!! !412034 7 61261!2 - 4"1!11 .; /!"0!12034. 124 0 I!7 < "!A!" H$%%>N 21 !412034. 2B!7 !" 03:21 8 I66 1!. 844. 2L2"1244. ! B20. /!"20< "6! !412034.2 20!/!4. 28"1024. 6""!. 80!4. F& 6! 0I !16. !412034. 2; B!7 ! 6@!44. !9621!/2"! 2B!7 6 I1 A.13 20!: 4. 4 0D; A 6 5:.2 ." I 45 ! 4"1 5L22 265 !" 03: 4. I664.2 "28"1 4 5:. "03 ! & "4 1 7 20!:B!! 2820521"5 12A 4!56!< A"0 0244.6! " 62"14.6 !" 03: 4!26 "!6 034. ./!"024!7 ! !412034 I B2; 4!4!5 @!A !A0!?244. 2@24!7< 1?2 !412034. ./!"024!7 "@!24!7 805 0/244. "074;2@24!7& I66< 20!:DL! !9621!/2"!2 2B!! 48 I4!/4.6! :4/24!56! !4120 < !" 03; :D1"5 40244.2 I024!5< /1 : 0521 8 A!13"5 I41! 44 I 0D/24!5 0/244.7 !4120 1 /4 I 2:0311 !9621!/2"! 2; B!7< 1?2 0/24!5 I41! 44 7 20!/!4. 1 /4 "1!& @!445 !9621! :4/!12034 7 622 :!"!1 1 14. 2; 0!:B!7< 1 .2 28 "10521 03: 120D O < C1 6 ! 4!"4!! B28 !9621!/2"! 2B!7 I41! 44 7 1 /4 "1! H" 4; 0244.6! I024!56!K "28"16! 5:. ." I 45 ! 8!1"5 ;
!: 8!13 4 60!:B!D /!"20< "8!I!< 2 22 0424!5< I024!5 /!"20& 2:0311. 4"1 5L27 I0. A0! 4. A 1 M,>< ,#< ,'< J$N&
! #$% &$'( )* $$'& + ,(-.
(
E
! ! /-0 )* 12/(2+ 344.%52+ 6%(*!
""6 1!6 A.4 244 2 8!99224B!034 2 424!2 " 4/034.; 6! :4/24!56!< :844.6! !82 !4120 du = f (u) dt
H$&$K
u(t0) ∈ U 0 = [u0, u0]
"13 94B!5 f (u) 422.4 4 42 1 6 1.1 6 !41202 G< " 82?L26 t0 ! 8 021 521 "0 !D (!@!B 4 :641 6 !41202 [t0, tk ] ⊂ G& A :4/!6 /22: U t - A: 64 ?2"1 U 0 6 6241 2624! t "!0 "!"126. H$&$K< 1 2"13
U t = u(t)|u(t0) ∈ U 0
H$&FK
4 ?2"1 U t6 ?4 4:13 "2/24!26 " 4 "1! 2@24!7< " 121"1; DL! :4/24!D U 0 ! 9!"! 44 6 2624! t&
"0! 28 0 ?!13< /1 . 0424 "0 !2 422.4 7 8!99224B!; 26 "1! 2@24!5 4/034.6 844.6< 1 805 "2/24!5 "2627"1 121 !7 6 ?4 ./!"0!13 80!4 0/DL2I "5 !4120S t ∂f ∂u(t) u(s)ds w(U ) = U − U = du = 0 0 | |du0 = 0 e t0 ∂u du0 [U,U ] [u ,u ] ∂u0 U t
t
t
H$&GK
82?12034 7 ! C1 6 50521"5 "14 :8/! B24!4!5 64 ?2; "1 Ut 0DA 7 6 6241 2624! t ! 282024!2 "1224! 1 /4 "1! C1 I B2; 4!4!5& 022 ""6 1!6 !412034.7 621 8< 1 .7 : 0521 "1 !13 " 85L!2"5 !412034.2 B24!< /1 6 ?21 "0?!13 !00D"1B!27 ; 451!5 " 8!6 "1! ! " 8!6 "1! 24! ! 4!?4! H!412034.K B24 "0/2 "!"126 424!7& "D8 6. 28 0I26< /1 48 "12/DL!6!"5 9 60 !412; 06! !: 851"5 " 121"1DL!2 !412034.2 2B!!< 1 6 /!"02 :51!2 !412034 I !412I0 M$N& 2826 4 !412034 6 "14"12 I 621! t"8 9 h(A, B) = max{|a − b|, |a − b|},
I82 4!?4!7 48/2 A :4/21 4!?4DD I4!B !4120< 24!7 48/224DD I4!B !4120& A :4/!6 /22: CI ([t0, tk ]) "14"1 422; .4. !412034 :4/4. 94B!7< 2820244. 4 !41202 [t0 , tk ] " 621! 7
hI (F, G) = maxt max f (t) − g(t) , f (t) − g(t) .
2 A:26 424!2 $&$ !412I034 6 424!D −L(t−t0 )
u(t) = u0e
I82
+
t t0
−L(t−s)
[f (u(s)) + Lu(s)]e
−L(t−t0 )
ds == u0e
g(u(s)) = e−L(t−s) [f (u(s)) + Lu(s)], L=
+
t t0
g(u(s))ds
H$&>K H$K
∂f max (u(t)) . t∈[t ,t ] ∂u 0 k
∂g (2I !8213< /1 !: 845 ∂u ≥ 0 805 0DA I t ∈ [t0, tk ]. 2@24!2 :8; /! H$&$K C!02414 1."4!D "2 422.4. 2@24!7 !412I034 I 424!5 H$&>K 1!< /1 u0 ∈ U 0 . 05 2@24!5 !412I034 I 424!5 H$&>K !" 03:26 !412034.7 62; 1 8 "028 12034. !A0!?24!7 H621 8 !K& 6226
V n+1(t) = U 0 e−L(t−t0) +
I82
t t0
G(V (n) (s))ds,
H$&'K
G(V (n) (s)) = [g(v (n) (s)), g(v(n) (s))], V (n) (s) = [v (n) (s), v(n) (s)], n = 1, 2, ...,
505D1"5 2"12"1244.6! !412034.6! "@!24!56! 94B!7 g(v), v < V (0) (s) = [u(0) e−L(s−t0 ) , u(0) e−L(s−t0 ) ], V (0) (s)-42 1 2 4/034 2 !A0!?24!2< " 82?L22 U t & ! C1 6 V (1) .A!21"5 1< /1 A. V (2) (s) ⊆ V (1)(s)< ! C1 6 6 I1 5!13"5 42 1 .2 I4!/24!5 4 :4/24!5 s& 1621!6 1?2< /1 !412034 2 "@!24!2 G(V (n) (s)) = [g(v(n) (s)), g(v(n) (s))]
8 021 521 4224"1 hI (G(V (n) , G(V (n+1) ) ≤ L hI (V (n) , V (n+1) ),
! 42 1 . 2"12"1244. "0 !5& 2"0 ?4.6! ""?824!56! 6. 6 ?26 :13< /1 621 8 $&' ; "14"12 I(R)) !6221 42 8!?4D 1 / V ∗ < 1 5 " 821 " U t ! 0DA. 26244. :4/24!5 t& 05 8 :1203"1 C1 I :621!6< /1 2; 1 P ic(V ) ≡ V 0 + tt0 G(V (s))ds 22 8!1 422.4.2 !412034 :4/4.2 94B!! 422.4.2 !412034 :4/4.2 94B!! ! 50521"5 "?!6D; L!6 621!2 hI ! 42 1 . 22 40 I& /2"12 C!02414 7 27 621!! 2820!6
h I (F, G) = max max e−L1(t−t0 ) (|f (t) − g(t)|, |f (t) − g(t)|) t∈[t0 ,tk ]
H$&,K
:82"3L1 ≥ L.
:2"14 < /1 A2 621!! hI ! h I A08D1 !4!414 "13D 14 "!1203; 4 "8!I& 62 1 I (n+1) h (t), V (n) (t)) = h I ( I (V
t t0
G(V (n) (s))ds,
t t0
G(V (n−1) (s))ds).
412034.7 !412I0 "!0 422.4 "1! 8.412I034 7 94B!! G :!@21"5 1 t t0
G(V (n) (s))ds = [
t t0
g(v(n) (s))ds,
t t0
g(v(n) (s))ds].
I8 !" 03:5 A :4/24!5 Ce = e−L1(t−t0)< Cs = eL1(s−t)e−L1(s−t0)< 0; /26 4224"1
hI (
t
G(V
(n)
(s))ds,
t
G(V (n−1) (s))ds) =
t0 t0 ⎞⎫ ⎛ t ⎪ t t t ⎬ ⎜ (n) (n−1) (n) (n−1) ⎠ )ds , g(v )ds − g(v )ds ⎟ max ⎪Ce max ⎝ g(v )ds − g(v ⎪ t ⎩ t ⎭ t t t 0 0 0 0 ⎧ ⎫⎫ ⎧ ⎪ ⎪ ⎪ t ⎨ t ⎬⎪ ⎬ ⎨ (n) (n) −L1 (t−t0 ) (n−1) (n−1) ds, L ds v ≤ max ⎪e max ⎪L v − v −v ⎪ t ⎩ t0 ⎩ ⎭⎪ ⎭ t0 ⎫ ⎧ ⎪ ⎪ ⎬ ⎨ t ≤ max ⎪L Cs max(|v (n) (s) − v (n−1)(s)|, |v(n) (s) − v (n−1) (s)|)ds⎪ t ⎩ ⎭ t0 t L (n) n (n−1) ≤ L max e−L1 (t−t0 ) dsh , V (n−1) ≤ 1 − e−L1 (tk −t0 ) h ). I (V I (V , V t t0 L1 L1 ≥ L k = L/L1 1 − e−L1(tk −t0) ⎧ ⎪ ⎨
" 03 6. .A0! < 1 ! !6221 62"1 1& 028 12034 0/!6 "0 !2 "?!626 "1!&
+K
"12"1244 < /1 4224"1 HF&>+K "280! ! 805 182034. 6 ; 4241 21 E(x)& 40 I!/4 28.8L!6 ""?824!56 6. B24!26 10 424!2 !: 84. 1 /4 I 2@24!5 ! "074;2@24!5< |yk (q) (x) − sqk (x)|, q = 1, . . . , r
! 0/26 y (q) (x) − s(q) (x) ≤
(exp( A −1) r−q h ω(h). (r + 1)! A
HF&>$K
8 A4.2 B24! A "4 .D1 " 8!6 "13 844 I 621 8 "1 24!5 "074; 94B!7< "!6!DL! 2@24!2 !" 84 7 :8/!< 4 "26 !4; 1202 [x0, xT ] ! @I2 "21! h < "1265L26"5 +& ". "1 7/! "1! 1 I 0I !16 6 I1 A.13 0/24. 42"0 ?4.6! ""?824!56!& 1< 6. 280 ?!0! 0I !16< 1 .7 "1 !1 !412034.2 B24! 64 ?2"1 1 /4. 2@24!7 :8/! HF&GGK 4 "4 2 " /214!5 8 @I S ; "028 12034 I 4 ?824!5 40!1!/2"! 9 60< :8DL! .?24!5< 1 .6 6 ?4 ./!"0!13 :4/24!5 "074;94B!7< "!6!D; L! 2@24!2< ! 282024!5 C1!6 9 606 !412034. "@!24!7< 0D/DL! 2@24!2& " 03 6. !:A2I26 ! C1 6 @I I !62; 424!5 !412034. 2B!7< 6. A 8!6 0!54!2 C9921 & 84; "0/2 B24! 2@24!7 "!"126 424!7 " 0!4274 7 7 /"13D< 6. 6 ?26 03: 13"5 "26 7 "074; 00 B!! " 6216! "074 H"12243D n ! 82921 6 d K< 8A!26.6! !: "0 !7 "1 7/! "1! ! 1 /; 4 "1! !A0!?24!7& O1 :6 ?4 "82013< " 03 6. 6226 4 8!13 40!1!/2"! :0 00 B!! 9 60. 805 2@24!7 "!"126 0!4274. 0I2A!/2"! 424!7< 28205DL! 621. "074;94B!7< /1 20!: 4 64 I! "!"126 "!6 034. ./!"024!7& "0! 42 A 8!6 4 8!13 B24! 64 ?2"1 2@24!7 "!"126 8!99224B!034. 424!7 " 420!4274 7 7 /"13D< "121 :8/ 0/!13 40!1!/2"! 9 60. 2@24!7 "!"126 420!4274. 0I2A!/2"! 424!7< /1 AL26 "0/2 42 :6 ?4 & C1 6 280 ?24 8 8< "4 44.7 4 "1 24!! "074; 94B!7 A 03@ I 82921< 64 I 6 4 ? 4 "26 621 8 58 27; 0 < 4 10!/!2 1 4! !62424!2 "074 : 0521 "L2"1244 ; "1!13 8 :1203"1 " 8!6 "1! !412034. B24 &
"0/2 C1 I 8 8 46 821"5 0/!13 9 60. 805 "074;94B!7< "!6!DL! 2@24!2< "2 :!B!! 94B!7< :!"5L! 1 4; /034. 844.& ! . 0424!! "028DL2I @I 844 I 621 8< "5:4; 4 I " ./!"024!26 !412034. "@!24!7< 805 4" :6 ?4 .A13 84 !: 8 4024!7S 4 8!13 !412034D B24 A0"1! :4/2; 4!7 21 4 -:4/4. 94B!7< "028 12034 2@5 2n C"12603; 4. :8/ 805 8!41 C1! 94B!7< 0!A 8 04!12034 4 ?8 6 @I2 !: 8!13 0!42!:B!D "!6 034. 9 60 C1! 2@24!7 4/034.6 :4/24!56& 022 6. "14 !6"5 4 C1 6 4024!! A 022 8 A4 & 1< "!6 034.2 9 60. "074;94B!7 sk (x)< "!6!DL! 6 4241. 2@24!7 :8/! HF&GGK - C1 "2 :!B!! 94B!7< :!"5L! 1 4/034. :4/24!7 y10 , . . . , yn0 805 0DA I k & 05 4" 8 A4 2"1! A ; :4/24!5 Φijk (y10 , y20, . . . , yn0 ) 805 !: 84. fk(j)< /"1DL! B2""2 ./!"024!5& 826 !: 8!13 0!42!:B!D C1! 94B!7 4 !41202 Y 0 < 505D; L!6"5 !:2824!26 !4120 Y10 × Y20 × . . . × Yn0& " 03:26 9 60 270 Φijk (y10, y20 , . . . , yn0 ) = Φijk (y1∗, y2∗ , . . . , yn∗ ) +
∂Φijk |∗ ∂y 0 z
(y10 − y1∗ ) + . . . HF&>FK
∂ 2 φijk 1 $n ∗ 0 ∗ ∗ ∗∗ l,s=1 2 ∂yl0 ∂ys0 |z (yl + θl (yl − yl ))(ys θl , θs ∈ [0, 1], z ∗ , z ∗∗ ∈ Y10 × . . . × Yn0
ijk ∗ 0 ∗ + ∂φ ∂y 0 |z (yn − yn ) + n
+ θs (ys0 − ys∗ )),
A :4/!6 /22: mk,i,j 0!4274D /"13 C1 I :0 ?24!5 mijk = Φijk (y1∗, y2∗ , . . . , yn∗ ) +
n !
∂Φijk 0 (yl − yl∗ ) 0 l=1 ∂y
HF&>GK
(2I !8213 "280! "13 4224"1 |Φijk (y10, y20 , . . . , yn0 ) − mijk (y10 , y20 , . . . , yn0 )| ≤ n n ∂ 2φijk 1 ∂ 2φijk ! 1 ! | 0 0 w(Yl0)w(Ys0 )| ≤ max l, s, Y 0 | 0 0 | w(Yl0 )w(Ys0 ) ≤ 2 l,s=0 ∂yl ∂ys 2 ∂yl ∂ys l,s=0
I82
∂ 2φijk 2 0 n(n − 1) n(n − 1) 1 0 maxl, s, Y 0 | 0 0 |w (Ymax) = Φw2(Ymax ), 2 ∂yl ∂ys 2 4 w(Yl0 ) = Yl0 − Yl0,
w(Y 0) = maxl w(Yl0 )
HF&>>K
∂ 2φijk | ∂yl0 ∂ys0 l, s = 1, . . . , n.
Φ = max l, s, Y 0 |
1 0I !16 :0D/21"5 "028DL26S (
• ?8 6 !41202 [xi−1, xi] 28"10526 .?24!5 fk j) 94;
B!!< :!"5L!2 1 03
1 4/034. 844.
• 126 2820526 ! 0!4274.2 /"1! " I0"4 9 602 HF&>GK • 8"10526 ! .?24!5< 1 .6 6. ./!"050! 6 4241.
"074; 94B!7
sk (x) = sk (xi−1) +
r ! j=0
− xi−1)j+1 (j + 1)!
HF&>#K
sk (x0) = yk0 , i = 1, . . . , n
HF&>'K
(x mkij (y10 , y20, . . . , yn0 ) k = 1, . . . , nr ,
B24!6 I2@4 "13 621 8 HF&>#K ! !" 03:26 ! C1 6 A :4/24!5 ek (x) = |yk (x) − sk (x)|& I8 0/26 (j) (r+1) $r yk (ξk,i−1 ) yk (xi−1 ) j (x − x ) + (x − i−1 j=0 j! (r+1)! j+1 $ i−1 ) sk (xi−1) − rj=0 mkij (y10, y20 , . . . , yn0 (x−x (j+1)! | ≤
|yk (x) − sk (x)| = |
|yk (xi−1 − sk (xi−1)| +
$r−1 (x−xi−1 )j+1 j=0 | (j+1)! |
xi−1)(r+1) −
HF&>,K
+ r+1
| i−1 |yk (r+1) (ξk ) − mkij (xi−1, s1(xi−1), . . . , sn (xi−1)) |x−x (r+1)! |
" 03: @!"3 4224"1 6 HF&>>K< 0/26 B24! 805 !: 84. (j+1)
|yk
(j+1)
(xi) − mkij (y10 , . . . , yn0 )| ≤ |yk
(j)
(xi) − fk (xi−1)|+
HF&>JK
n(n − 1)/4Φw2(Y0), k = 1, . . . , n, j = 1, . . . , r − 1,
40 I!/4
B24!26 (r+1)
|yk
(r+1)
(ξki ) − mkir (y10 , . . . , yn0 )| ≤ |yk
n(n − 1)/4Φw2(Y0)
(r)
(ξki) − fk (xi−1)|+
8"1055 0/244.2 B24! HF&>,K< !6226 |yk (x) − sk (x)| ≤ ek (xi) + (Lk (
n !
el (xi))
l=1 r−1 ! hj+1 n(n − 1) 2 Φw (Y0)) + + 4 j=0 (j + 1)! (r+1) ω(yk , h)
n(n − 1) hr+1 2 Φw (Y0)) ≤ ek (xi)(1+ HF&>%K + (Lk ( el (xi)) + 4 (r + 1)! l=1 n !
n ! n(n − 1) hr+1 hr+1 )( el (xi))(hLk e + ) + he Φw2(Y0)+ hLk e + (r + 1)! l=1 (r + 1)! 4
hr+1 hr+1 n(n − 1) (r+1) 2 ) Φw (Y0) ω(yk , h) ≤ ek (xi)(1 + ck h)+ (r + 1)! 4 (r + 1)! ck
I82
n ! l=1,
hr+1 b2 . el (xi) + hw (Y0)b1 + (r + 1)! l=k 2
1 n(n − 1) ), b1 = e Φ, (r + 1)! 4 n(n − 1) (r+1) Φw2 (Y0) + ω(yk , h), b2 = 4 k = 1, . . . , n
ck = Lk ((exp +
05 21 @!A E(x) 6 ?4 :!"13 4224"1 hr+1 b2B, E(x) ≤ (I + hA)E(xi) + hw (Y0 )b1B + (r + 1)! 2
I82 61!B A ! 21 B A.0! 282024. 422& I8 "0 ?! "028 12034 4224"1 hr+1 b2 + hw2 (Y0)b1 E(x) ≤ (1 + h A ) E(xi) + (r + 1)! (1 + h A ) E(xi) ≤ (1 + h A )2 E(xi−1 + hr+1 (1 + h A )( b2 + hw2 (Y0 )b1) (r + 1)! (1 + h A )2 E(xi−1) ≤ (1 + h A )3 E(xi−2 + hr+1 (1 + h A ) ( b2 + hw2 (Y0)b1), (r + 1)! 2
HF+K
················································································· (1 + h A )i E(x1) ≤ (1 + h A )i+1 E(x0 +
HF$K
hr+1 b2 + hw2 (Y0)b1), (1 + h A ) ( (r + 1)! i
! /!1.5< /1 ek (x0) = 0 805 "2 k 6. 0/26 E(x) ≤ (
i !
(1 + h A )l
HFFK
l=0 r+1
h exp A − 1 h + hw2 (Y0 )b1) ≤ ( + hw2 (Y0)b1). (r + 1)! A (r + 1)! r+1
224"1 HFFK :4/21 " 8!6 "13 280 ?244 I 621 8 "074; 00 B!! ! h → 0, w(Y0) → 0& "262 1!/2" 7 20!:B!! C1 ; I 0I !16 C1 ! 8!1 42 A 8!6 "1! :A!13 !4120 4/034. :4/24!7 4 60.2 8!4120. ! "028 12034 2@13 :8/! "1 ; 24!5 24! ! 4!?4! B24 805 :8/ " 0/244.6! 8!41206! 4/034. :4/24!7& ! 9!"! 44. :4/24!5 h, w(Y0) 24!2 ! 4!?4!2 B24! A81 0D/13 64 ?2"1 1 /4. 2@24!7< 10 424!2 C1 ; I 0D/24!5 6 ?4 B24!13 20!/!4< :!"5LD 1 @I ! !4120 4/034. :4/24!7& . 8.& C1 7 I02 ""61!D1"5 ".< "5:44.2 " "1 24!26 24! ! 4!?4! B24 64 ?2"1 2@24!7 "!"126 " !412034.6! 844.6! ! 10 424!7 C1! B24 1 I4!B " 4 "1! "2 1 /4. 2@24!7 ; :44 7 :8/!& ! C1 6 851"5 451!5 " 8!6 "1! "8 9 7 621; !2 AP28!4244 I !412034 I "@!24!5< " 121"1DL2I "1 24; 4 7 280I26 6 621 82 !412034 7 B242< " 4 "1! 1 /4. 2; @24!7< ! 420/@26 7 H 8!414 7K !412034 7 B24!& 8!6 "13 "8 9 7 621!2 AP28!4244 I "@!24!5 4 "!1 " 22 12 21!/2; "!7 12< 420/@26 "13 H 8!4145 A0!: "13K B24 !6221 1!/2" 2 :4/24!2< " 03 1?21 91 002034 "1! "56 ; 8!41 I4!B !4120 ! A0!: "13 ! " 4 "156 1 /4. 2@24!7& 412034.2 621 8. 4 851 !412034.2 94B!!< I41! 44 " 82?L!2 1 /4.2 2@24!5 "!"126 " !" 03: 4!26 :0!/4. 8; 8 & ?4 .820!13 0I !16.< "1 244.2 4 "4 2 8!99224B!03; 4. 4224"1 MFF< G$< GFN< 42/4 -:4 "14. "!6B!7 " /21 6 I; 4!B @!A M#< 'N< :0 ?24!5 58 270 H!0! 8I!2 58.K " B24 7 "11 /4 I /024 M$< F< $'< $J< F$< FFN< 2B!! 82921 MF'< FJN< !12B!5
! !0! 3D1 4 " 121"1DL26 94B! 4034 6 "14"12 MG,N< 1?2 ./!"024!7 B24 " !" 03: 4!26 C00!" !8 M$$< G+< >G< >>< >#N& AL!6 8 8 6 805 "2 C1! 621 8 50521"5 12A 4!2 4 ?824!5 0D/24!5 2@24!7 6 ?4 A 022 : I < 1?2 !" 03: 4!2 " 7"1 6 4 1 44 "1! 14 "!12034 0D/24!5& 232:4 2 I4!/24!2 !412034. 621 8 B24! 2@24!7 "!"126 8!9; 9224B!034. 424!7 "5:4 " 0!54!26 1 4:.26 I C9921 ; H < ! 85L2I C" 424B!034 6 "@!24!D @!!4. 0/26. B24 M$$< FFN& "4 4 7 !/!4 7 C1 I "0?!1 421 /4 "13 !A0!?24!5 !4120 6< 0D/26 I !6 64 ?2"1 2@24!7 ! "!034.7 "1< :/"1D C" 424B!034.7< C1 I 10 424!5& O1 6 ?4 AP5"4!13 1?2 20!/24!26 0!/2"1 424!7 "!"1262< " 121"1DL27 2; 824!D !412034. B24 < 1 2"13 "@!24!26 :624 "1! "14"1 1 6 6. 4 8!6 !".26D B24& 621!6< /1 !412034.2 621 8. 8D1 I41! 44.2 B24! I0 ; A034 7 @!A!< : 05DL!2 0/13 I0 A034 2 0D/24!2 1 /4. 2@2; 4!7& 412034.2 :4/24!5 6 I1 5!13"5 :8/2 0!A !412034 2 4/034 2 :4/24!2< 0!A !412034.7 621 7 /"1!& !2 !4120. A./4 28"105D1 42 2820244 "1! ! 42 A5:12034 505; D1"5 60.6! 20!/!46!& 8/!< 1 .2 0D/24. !4120.< 6 I1 ""61!13"5 0!A 424!5 H"!"126. 424!7K 48 !41206!< 0!A 621!: 44.2 "2627"1 424!7& C1 7 I02 !"4 621 8 "1 24!5 !412034. B24 < 1 .7 "; 4 4 4 28!12034 6 282024!! 9 60 !A0!?244. 2@24!7 40!1!/2" 6 !82< I41! 44 7 B242 I0 A034. @!A ! 4 ?; 824!D !412034. "@!24!7 H" 62"14 805 9 60 ! I0 A034. @!; A K& 2A 4!2 "1 !13 9 60. !A0!?244. 2@24!7 0DA 7 1 /2 t A0"1! G : 0521 A 13"5 " 0!54!26 C9921 < " 03 224 ; "!1 282024!2 !412034. "@!24!7 4 "0284!7 C1 0I !16< 1 2"13 "4!621 z @I "13z B24 64 ?2"1 2@24!7& ! C1 6 !6245; D1"5 " /4 - 0!4 6!034.2 94B!! H!0! "074;94B!!K :0!/4. "122427 I08 "1! ! 82921 < /1 AP5"4521"5 42 A 8!6 "13D . 04513 40!1!/2"!2 .08!& 621!6< /1 .A !8 "074;94B!7 :!; "!1 1 1 I < 805 ! "!"126 424!7 !: 851"5 B24!S 805 "!"126 424!7< !62DL! 0!4274D D /"13< !0! 420!4274D D /"13& 2:0311. 4"1 5L27 I0. A0! 4. A 1 M,'< ,,< ,J< ,%< J$< J'N&
( 8!
-'%7 %:%.7 '(%52+ %L'&( (#$2 %32+ (20$*!
EG&$& ". 20!:B!7 !412034. 2B!7 ! 2B!7 " 40244.6! I024!56! H8!46!/2" 7 1 /4 "13DK& !@26 "11 844.< : 05DL! 1 /4 28"10513 2L2"124; 4.2 /!"0& O1 9 6 28"1024!5 844. : 0!1 8!13 1 /4.2 .; /!"024!5 48 /!"06!< 0D/5 !9621!/2"!2 2B!! 48 B! 4034.; 6! /!"06!& ! C1 6 6 ?4 1 /4 ./!"0!13 :4/24!5 94B!7< !" 03:5 58. " A2" 42/4.6 /!"0 6 /024 & "13 R− "14"1 827"1!12034. /!"20< M− 42 1 2 2I 864 ; ?2"1 < 1 A?24!2 ϕ : R → M 4: 26 I024!26< 2"0! ϕ(x) = x 805 0DA I x ∈ M. I024!2 4:.21"5 6 4 1 44.6< 2"0! ϕ(x) ≤ ϕ(y) 805 "2 x, y ∈ R 1!< /1 x ≤ y; I024!2 4:.21"5 40244.6 4!: I024!26< 2"0! ϕ(x) ≤ x 805 "2 x ∈ R; I024!2 4:.21"5 40244.6 2 I024!26< 2"0! ϕ(x) ≥ x 805 "2 x ∈ R. !0 < 805 6 4 1 44. !241! 44. I024!7 !451 !"; 03: 13 "2B!034.2 A :4/24!5< 4!62< ϕup (x) ≥ x−805 6 4 1 44 ; I < 40244 I 2 I024!5< ϕlow (x) ≤ x−805 6 4 1 44 I < 4; 0244 I 4!: I024!5< ϕnear (x) = (ϕup(x) − ϕlow (x))/2− 805 I024!5 A0!?7@26 6@!44 ;28"1!6 6 /!"0& 0I !16. 6@!44. !9621!/2"! 2B!7 6 I1 A.13 28"102; 4. "261!/2"! "028DL26 !82S 82 6 :!B!5 248 < . 0424!2 2B!7 48 :12056! "1224! ! 8 A4.6! /"156!< I024!2< 4 ; 60!:B!5< 9 61 /!"20 " 0DL27 1 / 7& *1 A. !: 8!13 42 A 8!6.2 I024!5< !9621!/2"!2 2B!! . 045D1"5 " 1 /4 "13D< 2.@DL27 "14814D 805 !9621!/2" I "1 7"1 O < :126 2521"5 "11 & 05 C1 I .82024!5 8 A4.2 /"1! ""61!D1"5 B20.2 /!"0 ! 48 4!6! !: 851"5 2B!! .@244 7 1 /4 "1!& ! . 0424!! !9621!/2"! 2B!7 63D122 2L2"1244.2 /!"0 28"1024. /!"0 " 0DL27 1 / 7< !4120. 2L2"124; 4. /!"20 !4120. " I4!/4.6! :4/24!56! o /!"06! " 0DL27
1 / 7& ! "2 2B!5 48 /!"06! " 0DL27 1 / 7 !: 8!1"5 .A /2"12 2:0311 /!"20< A0!?7@! 426& ! C1 6 !A0!; ?24!2 /!"20 6 ?21 . 04513"5 865 " " A6!S " 6 L3D I024!5 A0!?7@26 6@!44 28"1!6 6 /!"0< " 6 L3D I024!5 A0!; ?7@26 6243@26 6@!44 6 /!"0& 05 !"4!5 803427@! 2B!7 6. A826 03: 13"5 "028DL!6! A :4/24!56!< A0!:!6! 1!6 844.6 ! 21 6 5:. "03& "13 !4120. 28"1024. C0262416!< !4802?L!6! 1! 84; 4. :!"3S
82"3 02 " 121"121 6243@27 I4!B2 !4120< 02 " 1;
21"121 A 03@27 I4!B2 !4120& "1 544.2 20!/!4.
C1 28"1024!5< 12/DL!2 4!6243@26 ! 4!A 03@26 0 ?!12034 ; 6 :4/24!D "28! /!"20 " 0DL27 1 / 7& 5024!5 /!"20 " 0DL27 1 / 7< . 85L! : I4!B. A0"1! 6@!44 28"1!6. /!"20< 28205D1"5 :4/24!56! 90I S
"0! !" 8!1 2.4!2 2B!!< 4!62< "!0 82024!5 4 0< 6. 0I26 "L2"1 4!2 B28.
2.21 2B!D " " AL24!26 A @!A2& 1< AL!7 0I !16 "0 ?24!5 !412034. 20!/!4 6 ?21 A.13 28; "1024 "028DL26 !82
!" "
#" "
40 I!/4 6 ?4 28"1!13 0I !16 ./!14!5 !412034. 20!; /!4< !412034 2 64 ?24!2 ! 82024!2 A81 10!/13"5 : "/21 8 04!; 12034 I 40!: I4!/4. 1 /2& ! 20!:B!! B28 805 2B!7 48 I4!/4.6! 1 /6! !412; 0 6. 8 0?4. A2"2/!13 !"1"1!2 1!0!1< 0/DL!6! A0!?7@! :844 6 6@!44 6 /!"0 20!/!4< ! A :4/26.
! & I8 B28< "L2"105DL5 "0 ?24!2 " 40244.6! I024!; 56! !621 !8
!
05 4"1! 4!5 0I !16 !412034. 2B!7 ! 2B!7 " 40244.6! I024!56! 2"36 C9921!4 !624513 " 14 @24!5< "5:.DL!2 2:0311. 6@!44. !9621!/2"! 2B!7 ! !9621!; /2"! 2B!7 I41! 44 7 1 /4 "1!< /"14 "1! 1 /4. 2B!7& A :4/!6 /22: ∗M − 6@!44D !9621!/2"D 2B!D< u, v− 4 ; 60!: 44.2 248.< 4!62< +M − 6@!44 2 "0 ?24!2< /M − 6@!4; 4 2 82024!2& I8 :6 ?4 8 :13 24"1< 40 I!/4.2 "028DL26 u + v = (u +M v) +M ((u −M u1 ) +M (v −M v1 ))
I82 u1 = (u +M v) −M v, v1 = (u+M ) −M u1, u + v - 1 /4.7 2:0311 2B!! "0 ?24!5& :1203"1 2I " 8!1"5 ""6 124!D :6 ?4. " 14 @24!7 62?; 8 :12056! "1224! 248 & 1< 6. !6226 9 60 805 "5:! 62?8 u + v ! u +M v 126!4 20!/!4< 1 .2 6 I1 A.13 ./!"024. " 6 L3D 2B!7 84 14 7 1 /4 "1!& C1 6 1 /4 "13 ./!"024!7 6 ?4 .@13< 40!5 ; /4.2 /024. ! "0 !!< /1 42 !" 8!1 22 0424!2& 28"1! 8 A4D /"13 /!"0 !62DL!6 .@244 2 0!/2"1 :5; 8 < 6 ?4 0/!13 0I !16. !9621!/2"! 2B!7 48 I4!/4.6! 1 /6! !4120 .@244 7 1 /4 "1!< 4!62< 8 74 7 1 /4 "1!& "13 x, y− /!"0< :!"44.2 O < z− 2:0311 6@!44 7 !962; 1!/2" 7 2B!!< !624244 7 4!6< zz− /4.7 /024& "26< /1 /!"0 x ! y 6 I1 A.13 :82024. 4 82 /"1!< I82 608@!2 :58. " ; 82?1"5 226244. xx, yy, A.13 6 ?21 4. 40D& I8 0I !16. !9621!/2"! 2B!7< " I41! 44 7 B24 7 1 /4 "1! 4 "4 2 .@24!5 0!/2"1 :58 805 28"1024!5 2:0311 !62D1 !8S "0 ?24!2 x + y = z + zz,
I82
z = x +M y, zz = w +M y, w = x −M z,
./!14!2 I82
x − y = z + zz, z = x −M y, zz = z1 − z2 ,
z1 = y −M w, z2 = v −M x w = z −M x, v = z −M w,
64 ?24!2 I82
x ∗ y = z + zz, z = p +M q, p = x ∗M y, q = x ∗M yy +M xx ∗M y, zz = p −M z +M q +M xx ∗M yy,
82024!2 I82
x/y = z + zz, z = p +M q, zz = p −M z +M q, p = x/M y, q = (x +M xx −M p ∗M yy)/M y.
20!:B!5 28"10244. 0I !16 124 126< /1 6. !" 03; :26 A./4.2 !9621!/2"!2 2B!!< 28"10244.2 A 03@!4"12 0; I !16!/2"! 5:. ! 2B! 44. "28 63D12< ! " 6 I12034.2 226244.2< !4802?L!2 "14814.6 1!6 844.& 02821 1621!13< /1 C9921!4.6 50521"5 8 8< "5:44.7 " !" 03: 4!26 4 !12; 05& !1203- C1 A0"13 651!< 8 "11 /45 805 28"1024!5 /!"20 " 0DL27 1 / 7 /!"20 " 9!"! 44 7 1 / 7& 0!4 4 !1205 2820521"5 6!4!6034 6 ! 6"!6034 6 :4; /24!D :1205 "1224! ! 80!42 641!"". /!"20& O1 20!/!4 4 !1205 2820521"5 1< /1 A. 6. 6 I0! 4!13 426 1 /4 2 :4/24!2 !:282; 4!5 8 6@!44. /!"20& AL2!:444.6 12A 4!26 50521"5 1 < /1 ; A. 6. 6 I0! 1?2 4!13 :4/24!5 "66. n− /!"20 ! !:2824!5 n− 624. 21 A2: 62?1 /4. 22 0424!7 651!& B28.< 1 . 20!:D1"5 827"1!5 48 !41206!< "4 .; D1"5 4 !"44. .@2 2B!5 " 40244.6! I024!56! ! 8!4; 6!/2" 7 1 /4 "13D& /!"0 B28< A 1DL! " !412034.6! /!"06!< 0D/24. ; B28.< . 045DL!2 !9621!/2"!2 2B!!< 0 I!/2"!2 2B!!< "2; B!034.2 !412034.2 2B!!< 1!2 AP28!424!2< 22"2/24!2 !4; 120 < ""1 54!2 62?8 !41206! ! 42 1 .2 8I!2& 4!6 14 ; "51"5 1?2 1!0!1. 805 4 ?824!5 !412034. "@!24!7 "1481; 4. 61261!/2"! 94B!7& 05 ?8 7 !: B28< AP28!4244. !4; 12927"4 7 I664 7 A 0 / 7< 124.6! 505D1"5 8 6 6241S
40!/!2 8 "11 /4 A 03@ I /!"0 621 < 05DL! 2824!; 26 2B!!< 1?2 41 03 : 5024!26 :6 ?4. @!A < 2D /2283 : . 8 6 2:0311 : I4!B. A0"1! 6@!44 -28"1!6. /!"20& "4 2 !:0 ?244. !4B! :A 14 4A B28< : 05D; L!7 C9921!4 2@13 ".< :4!DL!2 ! ./!"024!! !41203; 4. "@!24!7 9 60 !A0!?244. 2@24!7< 505DL!"5 4 "4 2 "!6 034. ./!"024!7&
E8!"! &
%%'0$/+ %L'&( 37 %+37 4&6 #12+ )*!
". :!6 827"1!5 "!6 034. ! ./!"0!12034. 0I !16 5; 05D1"5 ?4.6! !".26. 621 8 0/24!5 I41! 44. B2; 4 "!"126 A.4 244. 8!99224B!034. 424!7& !6 034.2 2; B!! !: 851"5 " 6 L3D "2B!0!:! 44. I664. "28"1< 5:. {|}~q|M,$< ,FN< kr{c| M$#N< rq{vM$JN ! 1 6 8 A4.< 1; ?2 I664. "!"126< 20!:DL! 4214.2 "!6 034.2 621 8.& ; 7 8 8 !0212024 " 27 40244 "13D 4 2@24!5 4214. " ! :8/< :6 ?4 "13D 0/!13 C9921!4.2 20!:B!!& ?4 1621!13 "028DL!2 ?4.2 :!6 827"1!5 "!6 034. ! .; /!"0!12034. 0I !16 S -40!: !" 84. "!"126 ! 8I 1 !62424!D 8 A4 I "!6 03; 4 I ! /!"0244 I 621 8 \ - "1 24!2 /!"0244 I 0I !16 " I0"4 :844.6 "0 !56\ -I9!/2"!7 !0! 1A0!/4.7 . 8 2:0311 & 1621!6< /1 !"44.7 :82"3 8 8 10!/21"5 1 !:2"14. 621 8 "!6 034 I 2@24!5 424!7 M'%< ,+< ,$N 126< /1 C1! 621 8 "1 51"5 1 /4.2 2@24!5 424!7< 28"10244. 621 8 805 B24!4!5 ; 42 !241! 44.2 "5:4.2 I9. A2: B!0 < :8DL!2 "074;94B!!< "!6!DL!2 1 /4 2 2@24!2 !" 84 7 "!"126. 424!7< ! 8022 !"; 03:26.2 805 4 ?824!5 !412034. "@!24!7 0/244.6 9 ; 606& 8 I9 6 4!621"5 64 ?2"1 2@!4 C ! " 4 "13 2 A: ; 4!7 α 4 4!& 28"1024!26 822 4:.21"5 " " A :!"! !49 6B!! 426< 84 :4/4 ! 04 "13D ""140!DL!7 "11 822 ! :; 05DL!7 ./!"0513 2I 12!"1!!& ?4 .820!13 AL!2 " " A. 28"1024!5 o 805 "2 I9 28"1024!5 ! "2B!9!/2"!2 o 805 82; 232& 28"1024!2 " 6 L3D 61!B. "62?4 "1! 50521"5 AL!6 805 I9 < 4 2820521 I9 84 :4/4 " 1 /4 "13D 8 !: 6 9!:6& "1 !4"1 6 C1 I 28"1024!5 50521"5 02I "13 0/24!5< 1?2 :6 ?4 "13 !" 03: 4!5 805 5:. < 8 "DL! A!1 D AA 1; 8 !/4. "028 12034 "127& 84 C1 28"1024!2 42 50521"5 "6.6 C 4 64.6 ! 18 26 "13 0I !16 < A 1DL! " 1!6 28; "1024!26< 42 6 ?21 A.13 4!?2 O(n2 )& 64!6< /1 61!B "62?4 "1! 2"13 8145 61!B :62 6 n × n 1 7 C026241 aij 2820521"5
"028DL!6 " 14 @24!26& ⎧ ⎨
aij = ⎩
1 2"0! 2@!4. xi, yj "62?4.2< 0 1!4 6 "0/2&
1< 8 4!2"034 7 "!"126 7 "!6 034. !0! 40!1!/2"! 2 A; : 4!7 !451 "/!113 I664D "!"126< : 05DLD 8!13 0I2A!/2"!2 2 A: 4!5 48 AP216! 8 "11 /4 AL27 6126; 1!/2" 6 "6."02 ! 8.& 4!< !0 0D/D1 "2A5 B28. 805 A 1. 40!1!/2" 6 !82 " 61!B6!< 805 2@24!5 0I2A!/2"! 424!7 ! "!"126 0!4274. 424!7< 805 I242B!! I66 /!"0244 ; I "/21< 805 ./!"024!5 "!6 034 6 !82 H I8 C1 :6 ?4 K !412I; 0 ! "66< 8!99224B! 4!5 ! 8I! :8/& 2"36 8 A4 ! C1 6< /1 "2 /!"0244.2 C99!B!241. 28"105D1"5 A./4 !82 B! 4034. 8 A27 " 42 I4!/244.6! 80!46! /!"0!1205 ! :46241205& /2"12 "4 4 7 "!"126. 805 20!:B!! 4@2I 0I !16 A.0 .; A4 "!"126 kr{c|M$%N& ?4 1621!13< /1 !" 03:26 2 0I !162 "!6 034 2 8!99224B! 4!2 42 "2I8 8 "21 C9921!4D 20!:; B!D< /1 12A21 "L2"1244. !:62424!7& 20 1 6< /1 < 15 "!6 03; 4 2 8!99224B! 4!2 50521"5 A: .6 21 6< 9 60 A 03@ 7 80!4. 184 0/!13 .?24!2 805 !: 84 7 !:;: "126!12034 I "1 62?1 /4. 2:0311 & :6 ?24 8I 7 8 8& "0! !6221"5 I66 4 6;0!A 5:. 40!1!/2"! ./!"024!7< 2:0311 6 . 0424!5 1 7 A821 !4122"DL22 4" .?24!2< 1 40!:! ; @I : @I 6 "2 21 . C1 7 I66.< 6 ?4 1 61!/2"! "I2; 42! 13 I66< ./!"05DLD 2I !: 84D A 022 C 4 64.6 A; : 6& 7 8 8 4:.21"5 1 61!/2"!6 8!99224B! 4!26& ! C1 6 :62 0/244 7 I66. 6 ?21 :13"5 42A 03@!6& 0I !162 !" 03: 4. :6 ?4 "1! C1! 8 8 8 & 1621!6< /1 :8/ ./!"024!5 !412034. "@!24!7 94B!7 A 03@ 2 :4/24!2 ! A2121 L2445 9 6 :!"! .?24!7 ! !2824!2 8 A4. /024 < " 03 C1 : 0521 "L2"1244 "4!:!13 18 26 "13 0I ; !16< 265 !" 0424!5& "13 Y = F (X1 , . . . , Xn) 28"10521 21 !"!4!5& (2D /"13 H 226244D Y K 4: 26 2820526 7 H bYaYb< bYjK "1 44!; 6! 226244.6! !: F < 1 .2 A826 "/!113 !" 03:26.6! HVdYbK& O1 6 ?21 A.13 28"1024 C0262414.6 I9 6 " 2@!46! Y, X1, . . . , Xn& 40 I!/4 6. "5?26 A0 < " "1 5L!7 !: m− 21 !"!4!5 " I9 6 Gm < :.5 " 14 @24!5 62?8 2820526.6! ! !" 03:26.;
6! 226244.6! ?8 6 21 2 ! ! 03 A0 2& *1 A. 2820!13< "L2"121 0! :!"!6 "13 62?8 226244.6! Xi ! Xj < 1 1 . :!"!1 84 ! 1 ?2 2262445 Y 46 4?4 "1 !13 :!"!6 "13
Xi
I82 Xj ⎧ ⎨ ⎩
d
⎧ ⎪ ⎪ ⎪ ⎨
Xj = ⎪ ⎪ ⎪ ⎩
Xi succ Xj Xj succ Xi Xi ptcom Xj
succ Xi ⇐⇒
∃Xk | (Xi
succ
Xi uses Xj or Xk ) and (Xj succ Xk )
! Xi ptcom Xj ⇐⇒ {∃Xk |(Xk succ Xi )and(Xk succ Xj )} B2"" I0 A034 I L24!5 6 ?21 A.13 28"1024 !82 0I !1; 6< 1 I !"!4!2 !" 8!1 4 82< "028 12034 "13 6 ; 8!9!B! 44. 21 6 ?4 0/!13 4 . 82& ⎧ Yi =
(t) Fi (X1, . . . , Xn )
⎪ ⎪ ⎪ ⎨
−→ (STATEGt −1 ) =⇒ ⎪ ⎪ ⎪ ⎩
Y1
Yi
(t+1)
= F1
(X1, . . . , Xn ) ... (t+1) = Fi (X1, . . . , Xn )
?8.7 @I 0I !16< A :4/244.7 46! R< " "1 !1 !: 9:. " :84!5 ! 6 8!9!B!! I9< : 1 7 "02821 9: !"0D/24!5 :!"!6 "1! "28! 226244.< !" 03:26. @I F1(t), . . . , Fi(t)& 9:2 " :84!5 I9 .820521"5 42" 03 40!:!26. "0/2< :!"5L! 1 1 I 50521"5 0! Yi 282024 ! !" 03: 4 .?24!! 4 844 6 @I2 0I !16 9:2 !"0D/24!5 :!"!6 "127 6 ?4 2820!13 @I! 2!S $& L2"121 0! (Xi , Xj )< 15< /1 Xi dXj & I8 `K Xi Yi & ⎧ ⎨
!62< ⎩
succ Xj 02/21< /1 Xj :624521"5 4 " 2 282024!2
X1 = 2X2 Yi = X1 X2
⎧ ⎨
=⇒ ⎩
X1 Yi
= 2X2 = 2X2X2
ZK Xi ptcom Xj 02/21< /1 Xj 1 ! Xj :624521"5 4 " 2 2; 82024!2 Yi & ⎧ ⎪ ⎪ ⎪ ⎨
!62< ⎪ ⎪ ⎪ ⎩
Yi
X1 = Z X2 = 3Z = X2 − X1
⎧ ⎪ ⎪ ⎪ ⎨
=⇒ ⎪ ⎪ ⎪ ⎩
X1 = Z X2 = 3Z Yi = 2Z
F& L2"121 Yj < 1 2< /1 Yi
succ Yj
and (∃Xk | (Xk
succ Yi) and (Xk
succ Yj ).
C1 6 "0/2 20!:21"5 "028 120345 :624 "2 4!: 22; 6244.< !624526. Yj , Yi< 4 ! 282024!5& ⎧ ⎪ ⎪ ⎪ ⎨
= Yl + Xn Yl = Yi !62< ⎪ ⎪ ⎪ ⎩ Y = X1 − Xn i ⎧ ⎪ ⎪ Yj = Xl ⎪ ⎨ =⇒ ⎪ Yl = X1 − Xn ⎪ ⎪ ⎩ Y = X1 − Xn i
⎧ ⎪ ⎪ ⎪ ⎨
Yj
=⇒ ⎪ ⎪ ⎪ ⎩
Yj = Yl + Xn Yl = X1 − Xn Yi = X1 − Xn
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M,+N k^Vddh`V r& qese}|S ` {|}~q| l`WX`[Y j^U d^_ha[ bhxYUYaRh`_ YV`Rh^ad
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& pR`R& cud& - $%J#& - ^_& >$< 3J& - c& J'#-J,'& M,GN 424 &&< !4 &&< I02 && !4B! "1 24!5 21 !412034. 2B!7 A& *!"0244.2 621 8. 624!! "0 @4 7 "28.&; "!A!"&; $%J+< 1&$$< 3#& - & $>,-$#G& M,>N !4 & &< I02 && 21 !412034. 2B!7 805 O O;' 2!41 3F>;J$& - "!A!"& - $%J$& FF "& M,#N !4 &&< I02 &&< 08@2 &t& " 03: 4!2 621 8 !4; 12034 I 40!: A0262 14" 1A2034 "1! I66 !; 0845 61261! ! 624!& - @241< $%J$& - 3',+& - & %$-%,& M,'N ! &&< I02 && A 84 6 621 82 2@24!5 A.4 244. 8!99224B!034. 424!7 " 4/034.6! 844.6! !82 !4120 49 6B! 44 ; 21!4.7 612!0 H!412034.7 40!:K& 2; !41 & - "4 5"< $%J%& - 3%& - & F%-G+& M,,N ! &&< I02 && ""028 4!2 !412034 I 621 8 "02; 8 12034. !A0!?24!7 49 6B! 44 ; 21!4.7 612!0 H!412034.7 40!:K& 2!41 &; "4 5"< $%J%& 3%& - & FJ-F%& M,JN ! &&< I02 && 0!54!2 C9921 z"1!z4 0/24!2 24! ! 4!?4! B24 2@24!7 "!"126 A.4 244. 8!99224B!; 034. 424!7 =& ./& 61& ! 61& 9!:& - $%%+& - & F%< 3$+&; & $#%G-$#%#& M,%N I02 && "1 24!2 " 85L!"5 24! ! 4!?4! B24 2@24!7 "!"126 A.4 244. 8!99224B!034. 424!7 12!0. "2; " D:4 7 49224B!! z1034.2 A026. !084 7 61261!!z& - 1 < $%%$& - & $G'-$>+&
MJ+N I02 && "1 44!2 /!"0244 ;40!1!/2"!2 621 8. B24! 2; @24!7 "!"126 " !412034.6! 844.6! "365 62?84 8; 45 @ 0;"26!4 z/2"12445 12 !5 8!99224B!034. 42; 4!7 I!8 8!46!! H!4"1!11 I!8 8!46!!< "!A!" ; "4 5; "!7 I "4!2"!121Kz& - "4 5"< $%%F& - & FF-F>& MJ$N {^[`_Y r&s& sVYUhW`_ kYR^bd j^U |aW_^dha[ p^_VRh^ad ^j eUbha`Uu }hxY; UYaRh`_ |V`Rh^ad ihR aRYU`_ }`R` rZdRU`WRd j^U `a aRYUa`Rh^a`_ q^ajYUYaWY ^a sVYUhW`_ ra`_udhd ihR rVR^`RhW {YdV_R YUhW`Rh^a& k`RY`RhWd< rll_hW`Rh^ad `ab p^jRi`UY& YZUV`Uu F#-k`W $< $%%G& ~ahYUdhRu ^j p^VR gYdRYUa ^Vhdh`a`< ~pr& - c& ,#-,,& MJFN {^[`_Y r&s& eVRYU `ab haaYU YdRh`RYd j^U dYRd ^j d^_VRh^ad ^j e}|d ihR haRYU`_ b`R` aRYUa`Rh^a`_ q^a[UYdd ^a q^lVRYU pudRYd `ab rll_hYb k`RY`RhWd& - pR&cYRYUZVU[ pR`RY ~ahYUdhRu< {Vddh`a ^W`_ rqk q`lRYU HpR& cYRYUZVU[K - $%%G& - c& %%-$++& MJGN ! &&< I02 && "1 24!2 " 85L!"5 24! ! 4!?4! B24 2@24!7 "!"126 A.4 244. 8!99224B!034. 424!7 =40& ./& 61& ! 61& 9!:& - $%%G& - & GG< 3F& - & F$%-FG$& MJ>N {^[`_Y r&s& elRh`_ VllYU `ab _^iYU Z^Vabd j^U dYRd ^j d^_VRh^ad ^j e}|d ihR haRYU`_ b`R` aRYUa`Rh^a`_ q^a[UYdd ^a aRYU`_ `ab q^lVRYU; r_[YZU`hW YR^bd ha pWhYaWY `ab |a[haYYUha[& - pR&cYRYUZVU[ pR`RY ~ahYUdhRu< aRYUa`Rh^a`_
^VUa`_ maRYU`_ q^lVR`Rh^adn< $%%>&; c& F+G-F+>& MJ#N I02 && ?824!2 1!6034. I41! 44. B24 64 ; ?2"1 2@24!7 "!"126 " !412034.6! 844.6! A& ./!"0!; 12034.2 124 0 I!!& - "!A!"< $%%#& - & >< 3$G& - & #J-'>& MJ'N {^[`_Y r_YYh s& p^_ha[ pudRYd ^j eUbha`Uu }hxYUYaRh`_ |V`Rh^ad ihR aRYU`_ }`R`S {h[^U^Vd `ab elRh`_ ]^Vabd krqprkk aRYUa`Rh^a`_ pul^dhV ^a pWhYaRhW q^lVRha[< q^lVRYU rUhRYRhW `ab `_hb`RYb sVYUhWd< pYlR& F'-F%< $%%#& - ]YU[hdWY ~ahYUdhR)`R Yd`R^WdWV__Y gVllYUR`_< `WZYUYhW k`RY`RhW Vab adRhRVR j)VU ra[Yi`abRY aj^U`RhW HYU`auK& - c& $$G-$$>&
C1 6 !0 ?24!! 6. ! 8!6 "!6 034.2 9 60. "074; "!6B!7 2@24!7< 1 .2 0/D1"5 "02 84 I < 8 @I !62424!5 4@2; I 621 8& ! "/212 I4!B 2@24!7 4 8 "11 /4 A 03@ 6 !41202 6. 0/26 9 60.< :4!6DL!2 ! :!"! ! 6513 $#+;F++ !0 ; A71& 1!/2"5 20!:B!5 621 8 :4/21 .: C1! 9 60 6 ; 80!< 1 .2 !: 851 "/21 :4/24!7 !412034. "@!24!7< /1 C9921!4 6 ?4 !:2"1! 805 I664. !4"16241 "!"12; 62 kr{c|< 1 ! 4 5:.2 ]^U_`ab c`dW`_& 2 6 "0/2 6. 826 4"1!26 "074; "!6B!! 805 "!"126. M$N< 4 !622 1 7 4I0584 !00D"1!21"5 0!54!2 C9921 805 A 03@!4"1 621 8 & @ 0I !16 821 420/@26.2 B24!< " A 84.2 1 0!54!5 C9921& 1< 6. ""61!26 "!"126 0!4274. 8!99224B!034. 42; 4!7 " "1 544.6! C99!B!2416! HG&$K
y = Ay,
I82 y− 21 :624 "1! 8< A− 61!B C99!B!241 ⎛
A=⎝
0 1 −1 0
⎞ ⎠,
" !412034.6! 4/034.6! 844.6! y01 ∈ [−0.1, 0.1], y02 ∈ [0.9, 1.1]. "02 2 I @I 0/26 805 6 4241 21 2@24!7 9 60.< :!"5L!2 1 03 1 4/034. :4/24!7 y01 , y02S 1
y01
2
y02
y (t) =
y (t) =
+
y02
−h
y02 h 3 h 4y01 h 5y02 h 2y01 − + + h− 2 6 24 120
y01
h 3 y01 y02 h 4 h 5 y01 h 2y02 + + − . − 2 6 24 120
"02 1 I @I ; 9 60.S y 1 (t) = −2 h 2y01 + y01 +
y 2 (t) = y02 −
4 y 2 h 3 31 h 6y01 2 h 4y01 4 h 5y02 + 2 y02 h + − 0 − − 3 15 3 360 h 7y02 13 h 8y01 h 9y02 h 10y01 + + − 45 2880 1440 14400
2 y 2 h 4 4 h 3y01 4 h 5y01 + 0 + − 2 h 2y02 − 2 h y01 + 15 3 3 8 2 9 1 13 h y0 h y0 h 7y01 31 h 6y02 h 10y02 + − − 2880 1440 45 360 14400
"02 12132I @I;9 60.S 9 y02 h 3 121 h 6y01 9 h 2y01 27 h 4y01 81 h 5y02 1 2 + y0 + + 3 y0 h + − − − y (t) = − 2 8 40 2 120 1
17 h 7y02 49 h 8y01 9 h 11y02 409 h 9y02 h 15y02 + − + − − 40 320 3200 8640 1728000 h 13y02 181 h 12y01 361 h 10y01 h 14y01 + + − 115200 12800 345600 28800 h 15y01 h 13y01 9 h 11y01 181 h 12y02 h 14y02 + − + + 115200 1728000 12800 3200 345600 5 1 2 4 81 h y0 27 y0 h 9 h 3y01 − + + 40 8 2 8 2 9 1 7 1 6 2 10 2 y y y y y0 9 h 2y02 409 h 17 h 121 h 361 h 49 h 0 0 0 0 − − 3 h y01 + − + − − 2 320 8640 40 120 28800 y 2 (t) = y02 −
1 6 "0/2 6. 0/!0! 2:0311. !62424!5 621 8 42" 03; !6 6 82034.6 "!"1266 424!7< !"44. 6 4 I9!! 826! *24 "3 &(& B24!4!2 9: I " "1 54!5 8!46!/2"! "!; "126& - &S < $%JJ& O1! 2:0311. !4122"4. 805 "424!5 " !"4; 4.6 6 4 I9!! 84!6 !: 8 8 B242 64 ?2"1 2@24!7 "!"126 A.4 244. 8!99224B!034. 424!7< " :844.6 1 6 6 4 I; 9!!& !2826 :82"3 9 60. 805 "074;94B!7< "!6!DL! 2@2; 4!5& 82"3 ""61!0"5 0!4274.7 0526.7 "B!0051 " 5:!6 12; 4!26 ! 402. 4/034. "0 !5 dy d2 y + ky = u, + α dt2 dt |u| ≤ 1, dy(0) y(0) = = 0. dt 82"3 y-"0545 2262445< u-"054 2 024!2< k, α- "1 544.2 C99!B!241.& 2: 12! AL4 "1! 6 ?4 "/!113< /1 C99!B!241 k !4!621 84 !: 12 :4/24!7 k ∈ {−1, 0, 1}\ "0/7 !: 034 I k
" 8!1"5 84 6 !: 12 : "/21 !:62424!5 6"@1A & . ! 8!6 "0/7< I8 k = 0, α = −1& . 8!6 A :4/24!5 y1 805 y ! y2 805 dy dt & "02 2 I @I 0/26 805 6 4241. y1 (t) 21 2@24!7 9 ; 60.< :!"5L!2 1 03 1 4/034. :4/24!7 y01 , y02 S ⎛
⎞
⎛
⎞
y02 ⎠ 5 ⎝ y01 y02 ⎠ 4 y01 1 ⎝ + + y (t) = h + h 120 360 24 44 ⎛ ⎞ ⎛ ⎞ 1 2 1 2 y y y y + ⎝ 0 + 0 ⎠ h3 + ⎝ 0 + 0 ⎠ h2 + 6 9 2 7 y01 h + y01
"02 1 I @I 805 6 4241. y1 (t) 21 2@24!7 - 9 60.S ⎛
⎞
⎛
⎞
⎛
⎞
y02 y01 ⎠ 10 ⎝ y01 y02 ⎠ 9 ⎝ 13 y01 13 y02 ⎠ 8 1 ⎝ + + + y (t) = h + h + h + 24400 16240 1440 1820 2880 1860 ⎛ ⎞ ⎛ ⎞ ⎛ ⎞ 1 2 2 1 1 2 y 31 y 6 y y 24 y 4 y 0 0 0 0 0 0 ⎝ ⎠ h6 + ⎝ ⎠ h5 + + ⎠ h7 + ⎝ + + 85 64 360 360 25 15 ⎛ ⎞ ⎛ ⎞ 1 2 1 2 2 y 4 y 2 y 8 y 0 0 0 0 ⎝ ⎠ h4 + ⎝ ⎠ h3 + 2 y1 + 3 y2 h2 + 2 y1 h + + + 0 0 0 3 3 7 3
y01 .
!28244.2 844 6 !0 ?24!! 9 60. "074; "!6B!7 2; @24!7 826 4"1!D1 B2"" ./!"024!7 .?24!7< !" 03:26. 46! 805 4 ?824!5 !412034. B24 64 ?2"1 2@24!7& O1! 9 60. !; A0!?D1 1?2 21 "8!I 121 !! 2@24!7 8!99224B!034. 424!7&
!0 ?24!2 F& 2:0311. "/21 ! I9!! !412034. B24 64 ; ?2"1 1 /4. 2@24!7& 05 "!"126. M$N< !"44 7 4!I2 < 1 D !" 03: 0! /2"12 12"1 7 1 . "2 621 8 !412034. B24 2@24!7 "!; "126 A.4 244. 8!99224B!034. 424!7< 6. 0/!0! 1!/2; "! 1 /4.2 ?8 7 8!412 21 2@24!7 B24!& 9: 7 0 "; "1! 64 ?2"1 2@24!7 "!"126. y = Ay, HG&FK I82 y− 21 :624 "1! 8< A− 61!B C99!B!241 ⎛
A=⎝
0 1 −1 0
⎞
⎠,
" !412034.6! 4/034.6! 844.6! y1 ∈ [−0.1, 0.1], y2 ∈ [0.9, 1.1], 28; "10521 8!?L!7"5 4B241!/2"!6 ?4 "156 81 ! ; /!DL!7"5 14 "!12034 8!414. "27& 05 A 03@!4"1 621 8 < I; 41!DL! 1 /4 2 0D/24!2 64 ?2"1 2@24!7< "1 4 C1 I 81 20!/!21"5 "02 k− A 1 e2πk :< 1 2"13 "02 2 I A 1 /1! '++ :& 4@26 "0/2 6. 0/26 1!/2"! 1 /4 2 0D/2; 4!2 /!DL2I "5 81 "02 84 I A 1< /1 !: A?24 4 !"42 $& 82"3 "! A"B!"" 10 ?24 y1 "! 8!41 y2 &
!"42 F !: A?24 2824!2 2@24!7 "!"126. "02 '+ A ; 1 < 8!?24!5 81 !: A?24. 4 "028 12034 "1! 0 " "127< " 0 ?244. 002034 8I 8I 4024!! "! & 621!6< /1 265 "/21 805 B24! 2@24!7 "02 $+++ A 1 9: 6 "14; "12 :4!621 G+ 6!41 805 ]k cq >J' ; py ! 62422 J 6!41 805 ]k cq >J' ; }y>&
!"42 G :4. B24! 2@24!7 "!"126. x = xy − x + R, y = −x2 − y,
!28244 7 A 12 !10354" I 9!:! }Y UY[^Uh^ M,FN& 1621!6< /1 2824!2 2@24!7 C1 7 "!"126. 12!:21"5 "028DL!6! " 7"16!&
"0! (x0, y0 -4/0345 1 /< 1 2@24!2 " 82?!1"5 "922 8!" r = max[R, (x20 + y02 )1/2]< I82 R > 0& (2I :13< /1 C1 "!"126 !6221 "2I8 1 03 84 "1B! 44 2 2@24!2< 1 2 "1 7/! & 2:0311. !412; 034 I B24!4!5 2@24!7< 0/244.2 }Y UY[^Uh^ M,FN ! :0!/4. :4/24!5 R< 826 4"1!D1 C" 424B!034.7 "1 I4!B C1! B24 & 0/244.2 4@26 621 82 B24! 2@24!7 C1 7 "!"126. ! 4/034 6 !41202 x ∈ [4.9, 5.1], y ∈ [4.4, 4.6] " 851"5 84 1 /< /1 02I !; 8213 4 !"42 G&
!"42 > 28"1024. I9!! 805 12 ?2 B24 < 4 " !: A?24!26 "! R< 40244 7 2&
!"42 # :4. B24! 2@24!7 "!"126. 4;82; 05 x = y, y = y − (1/3)y 3 − x,
:82"3 > 0 60.7 621& :2"14 < /1 427 "140!21"5 2! 8!/2"!7 2?!6- 02A4!5< !:; 2"14.2 8 4!624 4!26 20"B! 44.< 124 7 /21 7 1 . 40!/!2 /"1 < I82 121 !5 2: 624521 4024!2-20"B! 4.. /"1 & 05 1 I < /1 A. /!"0244 2820!13 20"B! 44.2 /"1! 1; 21 !7< 42 A 8!6 !412I! 13 "!"126 " @I 6 64 I 6243@2& 2; 824!2 !412034. B24 < 0/244. 4@26 0I !16 ! = 0.01 !2824. 4 !"42 #& 265 "/21 " "1!0 F+ "248&